proof that a finite abelian group has element with \delimiter⁢69640972⁢g⁢\delimiter⁢86418188=e⁢x⁢p(G)

Theorem 1

If G is a finite abelian group, then G has an element of order exp⁡(G).

Proof. Write exp⁡(G)=∏piki. Since exp⁡(G) is the least common multiple of the orders of each group element, it follows that for each i, there is an element whose order is a multiple of piki, say |ci|=ai⁢piki. Let di=ciai. Then |di|=piki. The di thus have pairwise relatively prime orders, and thus

|∏di|=∏|di|=exp⁡(G)

so that ∏di is the desired element.

Title

proof that a finite abelian group has element with \delimiter⁢69640972⁢g⁢\delimiter⁢86418188=e⁢x⁢p(G)